Derivative Calculator

This derivative calculator differentiates a function and simplifies the result. It handles polynomials, trigonometric, exponential and logarithmic functions, can go up to the third derivative, and can evaluate the slope at any point.

Result

f′(x) = 2x·sin(x) + x²·cos(x)

Function	f(x) = x²·sin(x)

Results update as you type.

How to Use the Derivative Calculator

Type the function, for example x^2 * sin(x) or exp(2x).
Pick the derivative: first, second or third.
Change the variable if your function uses t, u, y or z instead of x.
Optionally enter a point to evaluate f and its derivative there.

The result shows the derivative simplified, using ² ³ for powers and · for multiplication.

The Main Differentiation Rules

Rule	Function	Derivative
Constant	c	0
Power	xⁿ	n·xⁿ⁻¹
Constant multiple	c·f(x)	c·f′(x)
Sum	f + g	f′ + g′
Product	f·g	f′·g + f·g′
Quotient	f / g	(f′·g − f·g′) / g²
Chain	f(g(x))	f′(g(x))·g′(x)
Exponential	eˣ	eˣ
Natural log	ln x	1/x
Sine	sin x	cos x
Cosine	cos x	−sin x

Worked Examples

Power rule: f(x) = 3x⁴ − 2x + 7

Differentiate term by term:

3x⁴ → 3 · 4x³ = 12x³
−2x → −2
7 → 0

So f′(x) = 12x³ − 2.

Product rule: f(x) = x²·sin(x)

Let u = x² and v = sin(x), so u′ = 2x and v′ = cos(x).

f′(x) = u′v + uv′ = 2x·sin(x) + x²·cos(x)

Chain rule: f(x) = sin(x²)

The outer function is sin, the inner is x². The derivative of the outer is cos, applied to the inner, times the derivative of the inner:

f′(x) = cos(x²) · 2x = 2x·cos(x²)

Slope at a point: f(x) = x³ at x = 2

f′(x) = 3x², so f′(2) = 3 · 4 = 12. The tangent line at (2, 8) is y = 12x − 16.

Higher derivatives: f(x) = x³

f′(x) = 3x²
f″(x) = 6x
f‴(x) = 6

Tips and Common Mistakes

Forgetting the chain rule. The derivative of sin(3x) is 3cos(3x), not cos(3x). Whenever the inside isn’t just x, multiply by the inside’s derivative.
Treating a product like a sum. (x²·sin x)′ is not 2x·cos x. Use the product rule.
Mixing up log bases. Here, log means the natural log ln. The derivative of log10(x) is 1/(x·ln 10).
Degrees vs radians. Calculus formulas like (sin x)′ = cos x only hold when x is in radians.

To see a function next to its derivative, paste both into the graphing calculator. Where the derivative crosses zero, the original function has a peak, a trough or a flat point. To find those x values exactly, solve f′(x) = 0 in the algebra calculator.

Frequently Asked Questions

What is a derivative?

A derivative measures how fast a function is changing. Geometrically, f′(a) is the slope of the tangent line to the graph of f at x = a. If f(t) is the position of a car, f′(t) is its speed and f″(t) is its acceleration.

What is the power rule?

The derivative of xⁿ is n·xⁿ⁻¹. Bring the exponent down in front and subtract one from it. So the derivative of x³ is 3x², and the derivative of 5x⁴ is 20x³. A constant on its own has a derivative of zero.

When do I use the product rule and the chain rule?

Use the product rule when two functions are multiplied, (fg)′ = f′g + fg′. Use the chain rule when one function is inside another, (f(g(x)))′ = f′(g(x))·g′(x). For x²·sin(x) use the product rule; for sin(x²) use the chain rule.

How do I find the slope at a specific point?

Enter the function and type the x value in the "Evaluate at" box. The calculator shows both f(x) and the derivative at that point. For f(x) = x³ at x = 2, f(2) = 8 and f′(2) = 12, so the tangent line there has slope 12.

What does the second derivative tell you?

The second derivative is the rate of change of the slope. Where f″(x) is positive the graph curves upward (concave up); where it's negative it curves downward. It's also used to check whether a stationary point is a minimum or a maximum.

Which functions can I enter?

Powers with ^, sqrt, sin, cos, tan, sec, csc, cot, their inverses (asin, acos, atan), sinh, cosh, tanh, exp and log (natural log), combined with +, −, * and /. Log is the natural logarithm; use log10 for base 10.

Sources & Assumptions

Last reviewed: October 26, 2026